High-resolution real-time Fourier transform based on optical frequency comb injected frequency shifting loop

Weiqiang Lyu; Weiqiang Lyu; Zhengkai Li; Zhengkai Li; Lingjie Zhang; Lingjie Zhang; Zhen Zeng; Zhen Zeng; Zhiyao Zhang; Zhiyao Zhang; Shangjian Zhang; Shangjian Zhang; Yong Liu; Yong Liu

doi:10.1364/OE.507151

1. Introduction

Real-time spectrum analysis is a key functional module for capturing transient signals in modern electronic reconnaissance systems, which is generally achieved by using analog-to-digital conversion and fast Fourier transform (FFT) in the digital domain [1]. For capturing a broadband signal or a signal with its center frequency up to tens of GHz, an ultra-high-speed broadband analog-to-digital converter (ADC) must be used, which brings a huge computational pressure to the backend digital signal processing (DSP). Photonic-assisted real-time Fourier transform, which maps the signal spectrum to the time domain in advance, is a powerful tool to facilitate directly obtaining the signal spectrum through analog-to-digital conversion [2–8]. Hence, the large time delay induced by DSP can be avoided.

Photonic-assisted real-time Fourier transform is generally achieved by using a large amount of group-velocity dispersion (GVD) to map the signal spectrum to the time domain. The requirement for GVD is to meet the far-field condition, i.e., |ΔT²/2Ф| << 1, where ΔT and Ф are the signal duration and the amount of GVD, respectively. As a result, to realize real-time spectrum analysis of a signal with a bandwidth at gigahertz level, the GVD is required to be far larger than 0.5 ns², which is equal to that in a spool of standard single-mode fiber (SMF) with a length far larger than 2.2 × 10⁴km at 1550 nm. The transmission time delay far larger than 110 ms greatly reduces the real-time performance. On the other hand, limited by the practical amount of GVD provided by the optical dispersive mediums, including optical fibers and chirped fiber Bragg gratings, the frequency resolution is typically beyond a few gigahertz. To circumvent the requirement for a huge amount of GVD, time lens, i.e., a quadratic temporal phase modulation similar to the function of a spatial thin lens in Fraunhofer diffraction, has been employed to enhance the frequency resolution of the photonic-assisted real-time Fourier transform [9–11]. Nevertheless, due to the limited time aperture, the frequency resolution, which is inversely proportional to the time window of the real-time Fourier transform, is confined to hundreds of megahertz. Schemes based on constructing a large equivalent GVD have also been proposed to circumvent the use of ultra-long optical fibers [12–14]. Based on the spectrally-discrete dispersion implemented by using a 0.5-m-long optical fiber loop, real-time Fourier transform with an unambiguous bandwidth of 400 MHz and a frequency resolution of 25 MHz has been realized, where the equivalent GVD is equal to that in a spool of SMF with a length of about 4.6 × 10⁴km [12]. To enhance the unambiguous bandwidth, a bandwidth slicing technique, which is similar to the channelization architecture, has been adopted [13]. Besides, bandwidth magnification method has been employed to equivalently amplify the GVD [14]. The performance of a real-time Fourier transform system can be quantified by using the time-bandwidth product (TBWP), i.e., the ratio of the unambiguous bandwidth to the frequency resolution. The spectrally-discrete-dispersion-based real-time Fourier transform scheme is with a TBWP of 16, which is unable to analyze time-variant signals with large TBWPs, e.g., linearly chirped signals.

In recent years, frequency shifting loop (FSL) has been proven to be a powerful tool to realize spectrum analysis [15], fractional Fourier transformation [16], chirped waveform generation [17], spectral shaping [18], photonic correlation [19], absorption spectroscopy [20,21], etc. The output of the FSL is composed of multiple signal replicas simultaneously shifted in the temporal domain and the frequency domain, which is equivalent to experiencing a large dispersion. For an FSL with a round-trip time of 100 ns and a frequency shift of 80 MHz, the equivalent GVD is equal to 1.25 ns/MHz, which corresponds to that in a spool of SMF with a length of 9.18 × 10⁶km. Most importantly, the equivalent GVD can be varied in a large range through simply changing the frequency shift. The unambiguous bandwidth of the FSL is equal to its free spectral range (FSR), and the frequency resolution is equal to the ratio of the FSR to the number of round trips. Hence, the TBWP of the FSL-based real-time Fourier transform system is equal to the number of round trips, which generally exceeds a few hundred, e.g., above 400 in [15]. Nevertheless, due to the gain saturation effect and the cumulative amplified spontaneous emission (ASE) noise in the FSL, it is extremely difficult to further increase the number of round trips [22]. Hence, the frequency resolution is still limited.

In this paper, an FSL-based high-resolution real-time Fourier transform scheme is proposed and demonstrated based on optical frequency comb (OFC) injection. Through setting the frequency interval between neighboring teeth in the coherent OFC to be equal to an integer multiple of the frequency shift and also the FSR of the FSL, the number of the effective output signal replicas will be increased by M times, where M is the tooth number of the optical frequency comb. On this condition, the output optical pulse corresponding to a single-tone frequency will be narrowed by M times, which enhances the frequency resolution in the real-time spectrum analysis. In the experiment, the output optical pulse corresponding to a single-tone frequency is narrowed by three times through using a coherent three-tone optical carrier. The frequency resolution is enhanced from 60 kHz to 20 kHz.

2. Operation principle

Figure 1 shows the proposed high-resolution real-time Fourier transform scheme. Firstly, the signal under test (SUT) is loaded onto a coherent OFC with M teeth by using an electro-optic Mach-Zehnder modulator (MZM). On this condition, the SUT is multicast by each tooth of the OFC. Then, the multiple optically-carried SUT replicas are injected into an FSL consisting of a 2 × 2 optical coupler to construct a loop with input and output ports, an acousto-optic frequency shifter (AOFS) to achieve frequency shift, an optical amplifier to compensate for the loop loss, and an optical bandpass filter (OBPF) to filter out the out-of-band ASE noise and control the number of round trips N. In the FSL, after each round trip, the optically-carried SUT replicas are frequency shifted by f_s and delayed by τ_c. When f_s is set to be equal to k/τ_c (k is a positive integer), a huge equivalent GVD is constructed, which realizes real-time Fourier transform of the injected optically-carried SUT replicas. For each injected optically-carried SUT replica, the output temporal waveform of the FSL maps the spectrum of the injected SUT as that in [15], where the unambiguous bandwidth is equal to the FSR of the FSL, i.e., 1/τ_c, and the frequency resolution is equal to the ratio of the FSR to the number of round trips, i.e., 1/(Nτ_c). In the proposed scheme, the effective number of round trips can be greatly enhanced through setting the frequency interval between neighboring teeth in the coherent OFC to be equal to an integer multiple of the frequency shift, i.e., f_rep = (N + 1)f_s, as shown in Fig. 1(b). In such a case, the number of the signal replicas from the FSL is increased by M times, compared with the situation by using a single-tone optical carrier. Since the OFC is highly coherent, the signal replicas from the FSL are phase-locked. It is equivalent to that the effective number of round trips is enhanced by M times, i.e., M × N, which leads to M times pulse width reduction of the generated optical pulse from the FSL. Hence, the frequency resolution is improved by M times.

Fig. 1. Schematic diagram of the high-resolution photonic real-time Fourier transform scheme. OFC: optical frequency comb; MZM: Mach-Zehnder modulator; SUT: signal under test; OC: optical coupler; AOFS: acousto-optic frequency shifter; OA: optical amplifier; OBPF: optical bandpass filter; RF: radio-frequency; PD: photodetector; OSC: oscilloscope.

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Mathematically, when a single optically-carried SUT is injected into the FSL, the output is the superposition of the recirculating SUT replicas. The optical field can be written as

(1)$${E_{out}}(t )= {E_{in}}{e^{ - j{\omega _0}t}}\sum\limits_{n = 0}^N {H({{\omega_0} + n{\omega_s}} )} {e^{ - jn{\omega _s}t}}{e^{jn{\omega _\textrm{0}}{\tau _c}}}{e^{j\frac{{{\omega _s}{\tau _c}}}{2}n({n + 1} )}}$$

where E_in and ω₀ are the amplitude and the center angular frequency of the injected optically-carried SUT. The real positive function H(·) characterizes the amplitude envelope of the SUT replicas from the FSL, which is determined by the net gain and the frequency response of the FSL. ω_s is the angular frequency shift, i.e., ω_s = 2πf_s. When the frequency shift is set to be an integral multiple of the FSR, i.e., f_s = k/τ_c, the phase term (ω_sτ_c/2)×n(n + 1) = kn(n + 1)π is an integral multiple of 2π. Hence, Eq. (1) can be rewritten as

(2)$${E_{out}}(t )= {E_{in}}{e^{ - j{\omega _0}t}}\sum\limits_{n = 0}^N {H({{\omega_0} + n{\omega_s}} )} {e^{ - jn{\omega _s}\left( {t - \frac{{{\omega_\textrm{0}}{\tau_c}}}{{{\omega_s}}}} \right)}}$$

Considering the Poisson summation formula [23]

(3)$$\sum\limits_{n ={-} \infty }^{ + \infty } {H({{\omega_0} + n{\omega_s}} )} {e^{ - jn{\omega _s}t}} \propto \sum\limits_{n ={-} \infty }^{ + \infty } {h\left( {t - n\frac{{2\pi }}{{{\omega_s}}}} \right)}$$

where h(t) = ∫H(ω)e^-jωtdω. Since H(ω₀+ nω_s) = 0 for n < 0 and n > N in Eq. (2), $\mathop \sum \limits_{\textrm{n = 0}}^\textrm{N} \textrm{H}({{\mathrm{\omega }_\textrm{0}}\textrm{ + n}{\mathrm{\omega }_\textrm{s}}} ){\textrm{e}^{\textrm{ - jn}{\mathrm{\omega }_\textrm{s}}\textrm{t}}}\textrm{ = }\mathop \sum \limits_{\textrm{n ={-} }\infty }^{\textrm{ + }\infty } \textrm{H}({{\mathrm{\omega }_\textrm{0}}\textrm{ + n}{\mathrm{\omega }_\textrm{s}}} ){\textrm{e}^{\textrm{ - jn}{\mathrm{\omega }_\textrm{s}}\textrm{t}}}$. Then, Eq. (2) can be simplified as

(4)$${E_{out}}(t )\propto {E_{in}}{e^{ - j{\omega _0}t}}\sum\limits_{n ={-} \infty }^{ + \infty } {h\left( {t - \frac{{{\omega_\textrm{0}}}}{{{\omega_s}}}{\tau_c} - n\frac{{2\pi }}{{{\omega_s}}}} \right)}$$

Since the envelope of H(ω) is generally a quasi-rectangle function with a spectral bandwidth of Nf_s, h(t) is a sharp pulse with a pulse width of about 1/(Nf_s) at t = 0. Therefore, Eq. (4) indicates that the output temporal waveform of the FSL injected by a single angular frequency ω₀ is a periodic pulse train with a repetition frequency of f_s. Besides, there is a relative time delay determined by the injected signal frequency as

(5)$$\Delta t({{\omega_\textrm{0}}} )= \frac{{{\omega _\textrm{0}}}}{{{\omega _s}}}{\tau _c}$$

which is the critical feature used for realizing real-time Fourier transform.

Two injected signals with a frequency interval equal to the FSR, i.e., f_c, will lead to an identical temporal waveform at the output of the FSL since the relative time delay between them is equal to the period of the generated pulse train, i.e., 1/f_s. In other words, the unambiguous bandwidth of the FSL-based real-time Fourier transform scheme is equal to the FSR of the FSL, i.e., f_c. Moreover, a single-tone optically-carried SUT seeded into the FSL will be mapped to an optical pulse train with a pulse width of δt = 1/(Nf_s), as shown in Eq. (4), which exhibits the intrinsic frequency resolution of the FSL-based real-time Fourier transform. Combined with Eq. (5), the frequency resolution can be expressed as

(6)$$\delta f = {f_s}{f_c}\delta t = \frac{{{f_c}}}{N}$$

It can be seen from Eq. (6) that the frequency resolution is essentially limited by the number of round trips, i.e., N.

In the proposed scheme, multiple coherent optically-carried SUT replicas are injected into the FSL to increase the number of recirculating optically-carried SUT replicas from the FSL, which equivalently enhances the number of round trips. In this circumstance, the output optical field of the FSL can be written as

(7)$${E_{out}}(t )\propto {E_{in}}\sum\limits_{m = 0}^{M - 1} {{e^{ - j({{\omega_0} + m{\omega_{rep}}} )t}}} \sum\limits_{n ={-} \infty }^{ + \infty } {h\left( {t - \frac{{{\omega_\textrm{0}}}}{{{\omega_s}}}{\tau_c} - n\frac{{2\pi }}{{{\omega_s}}}} \right)}$$

where ω_rep = 2πf_rep. Since the frequency interval of the multiple optically-carried SUT replicas is equal to an integral multiple of the frequency shift and also the FSR of the FSL, i.e., f_rep = (N + 1)f_s = k(N + 1)f_c, the time delay between the pulse trains corresponding to the first and the m-th optically-carried SUT replicas is equal to mf_repτ_c/f_s = mk(N + 1)/f_s, which is an integral multiple of the period of the generated pulse train. Hence, each of the injected optically-carried SUT replicas will produce an identical frequency-to-time mapping (FTM) pulse train after passing through the FSL. In addition, the carrier frequency interval of the multiple FTM pulse trains mapped by the multiple optically-carried SUT replicas, i.e., f_rep, is consistent with that of the injected multiple optically-carried SUT replicas. The OFC used to replicate the SUT is phase-locked, which leads to a phase-locked relationship between the generated multiple FTM pulse trains. Hence, the coherent superposition will arise among them, which results in a M times reduction in the pulse width of the generated FTM pulse train, i.e., from 1/(Nf_s) to 1/(MNf_s). Therefore, the frequency resolution of the FSL-based real-time Fourier transform is enhanced by M times. It should be pointed out that the GVD in the SMF may introduce a temporal offset to the pulse train generated by each optically-carried SUT replica, which has not been considered in the theoretical analysis. Nevertheless, since the length of the SMF in the FSL is very short, the GVD-induced temporal offset is negligible for a small bandwidth of the coherent OFC. On this condition, the GVD effect has no influence on the frequency resolution improvement of the proposed scheme. In addition, the influence of the GVD effect can be eliminated through using a coherent OFC near 1310 nm, where the GVD coefficient for the SMF is almost equal to zero.

3. Experimental results and discussion

An experiment is carried out to demonstrate the proposed scheme to realize high-resolution spectrum analysis. In the experiment, the FSL is operated near 1550 nm and with a maximum frequency interval of the injected signal replicas equal to 16 GHz, where the influence of the GVD can be ignored. Firstly, a three-tone optical carrier used as a coherent OFC is generated through injecting a continuous-wave (CW) light from an ultra-narrow linewidth laser source (NKT Photonics BASIK X15) into an MZM (Fujitsu FTM7938EZ) biased at its low bias point and driven by an 8-GHz radio-frequency (RF) signal from a microwave source (Rohde&Schwarz SMB 100A) as shown in Fig. 2. Mathematically, the generated coherent three-tone optical carrier can be expressed as

(8)$${E_{OFC}}(t )= {E_0}{e^{i{\omega _0}t}}\left( {\cos \left( {\frac{{{\varphi_b}}}{2}} \right){J_0}\left( {\frac{m}{2}} \right) - \sin \left( {\frac{{{\varphi_b}}}{2}} \right){J_1}\left( {\frac{m}{2}} \right)({{e^{i{\omega_{rep}}t}} + {e^{ - i{\omega_{rep}}t}}} )} \right)$$

where E₀ and ω₀ are the amplitude and the angular frequency of the CW light, respectively. φ_b and m are the direct-current (DC) bias-induced phase shift and the modulation index of the MZM, respectively. J_n(·) represents the n^th-order Bessel function of the first kind. To ensure that each tooth of the generated three-tone optical carrier is with an identical phase and power, and that the high-order sidebands are suppressed to an extremely low level, φ_b = 1.1π and m = 0.2π are set in the experiment.

Fig. 2. Schematic diagram of generating a coherent three-tone optical carrier. LD: laser diode; RF: radio-frequency; DC: direct-current; MZM: Mach-Zehnder modulator.

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The SUT is multicast by the three-tone optical carrier via using another MZM (Sumicem T.MXH1.5-20PD-ADC-LV-001). The generated optically-carried SUT replicas are injected into the FSL with an FSR of 5 MHz via a 2 × 2 optical coupler with a power splitting ratio of 50:50 to achieve real-time Fourier transform. In the FSL, an AOFS (AA Opto-Electronic MT80-B10-IIR30-Fio-SM0) driven by a single-tone RF signal from a microwave source (Rohde&Schwarz SMB 100A) is employed to achieve a frequency shift of 80 MHz in each round trip, where the 80 MHz frequency shift is 16 times of the FSR to guarantee the realization of real-time Fourier transform. The net gain of the FSL is set to be slightly less than 0 dB by using a gain-tunable erbium-doped fiber amplifier (EDFA, Amonics AEDFA-IL-23-B-FA). A programmable OBPF (II-VI WaveShaper 4000B) is inserted into the FSL to filter out the out-of-band ASE noise and to avoid parasitic loop oscillation. In addition, the polarization of the intracavity optical field is adjusted by using a polarization controller to achieve optimal interference effect since the pigtails of the optical devices in the FSL are made of SMFs. A fraction of the intracavity optical field is extracted out from the FSL via the 2 × 2 optical coupler. The output optical signal is converted to an electrical signal by utilizing a high-speed photodetector (Discovery DSC20H) with a 3-dB bandwidth of 33 GHz. Finally, the electrical FTM waveform is captured by using a real-time oscilloscope (Tektronix DPO75002SX) with a sampling rate of 100 GSa/s and an analog input bandwidth of 33 GHz.

Figure 3 shows the optical spectra of the CW light from the laser diode and the generated coherent three-tone optical carrier. It can be seen from Fig. 3(b) that each tone of the optical carrier has an almost identical optical power, and the high-order modulation sideband suppression ratio is up to 32 dB. In the experiment, the maximum bandwidth of the FSL output injected by a single-tone optical carrier is about 8 GHz. Hence, the frequency interval between neighboring tone of the three-tone optical carrier is set to be 8 GHz to ensure that it is equal to an integer multiple of the frequency shift and also the FSR of the FSL.

Fig. 3. Optical spectra of (a) the CW light and (b) the generated coherent three-tone optical carrier.

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Firstly, the CW light from the laser diode and the generated coherent three-tone optical carrier are separately injected into the FSL. The bandwidth of the OBPF is optimized to obtain a nearly flat-top optical spectrum and a narrowest pulse width from the FSL, which guarantees a high signal-to-noise ratio (SNR) of the generated optical pulse train simultaneously. Figure 4(a) and (b) present the optical spectrum and the temporal waveform from the FSL under CW light injection, respectively. Figure 4(c) and (d) exhibit the optical spectrum and the temporal waveform from the FSL under three-tone optical carrier injection, respectively. It can be seen from Fig. 4(a) and (c) that the output spectral bandwidth of the FSL injected by the three-tone optical carrier is increased by almost three times, compared with that injected by the CW light. Accordingly, the generated FTM pulse is narrowed by three times, i.e., from 150 ps to 50 ps, as shown in the insets of Fig. 4(b) and (d). This leads to a frequency resolution enhancement from 60 kHz to 20 kHz based on Eq. (6), where N is equal to 83.

Fig. 4. Experimental results for the FSL injected by the CW light and the coherent three-tone optical carrier, respectively. (a) Output optical spectrum under CW light injection. (b) Output temporal waveform under CW light injection. (c) Output optical spectrum under three-tone optical carrier injection. (d) Output temporal waveform under three-tone optical carrier injection.

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Then, single-tone RF signals at 29.95 MHz, 30 MHz and 30.05 MHz are separately loaded onto the CW light and the three-tone optical carrier via electro-optic modulation. The modulated optical signals are injected into the FSL to realize real-time Fourier transform. Figure 5(a) and (b) show the temporal waveforms from the FSL under CW light injection and three-tone optical carrier injection, respectively. Since the frequency interval of the RF signals is 50 kHz, they cannot be distinguished by using the FSL under CW light injection due to the limited frequency resolution of 60 kHz, as shown in Fig. 5(a). However, these signals can be clearly distinguished without any temporal overlap under three-tone optical carrier injection as shown in Fig. 5(b), indicating effective frequency resolution enhancement based on the proposed scheme is achieved.

Fig. 5. Temporal waveforms from the FSL under (a) CW light injection and (b) three-tone optical carrier injection.

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The proposed scheme is also demonstrated through simultaneously analyzing three single-tone RF signals. Figure 6(a) shows the spectrum of the input three single-tone RF signals at 29.95 MHz, 30 MHz and 30.05 MHz, which is generated by using an arbitrary waveform generator (RIGOL DG5352). Figure 6(b) and (c) exhibit the temporal waveforms from the FSL under CW light injection and three-tone optical carrier injection, respectively. It can be seen from Fig. 6(b) that the pulses corresponding to the three single-tone RF signals are severely overlapped with each other. If any two neighboring RF signals are with a large power difference, the generated FTM pulses corresponding to the RF signal with a lower power will be buried in the envelope of that corresponding to the RF signal with a higher power. This will result in the loss of the weak signal during the spectrum analysis. In Fig. 6(c), the FTM pulses corresponding to the three single-tone RF signals are without any temporal overlap, which is beneficial for recognizing two signals with very close frequencies but a large power difference.

Fig. 6. Experimental results for simultaneously analyzing three single-tone RF signals. (a) Spectrum of the input three single-tone RF signals at 29.95 MHz, 30 MHz and 30.05 MHz. (b) Temporal waveforms from the FSL under CW light injection. (c) Temporal waveforms from the FSL under three-tone optical carrier injection.

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Finally, the FTM linearity of the proposed scheme is characterized through analyzing five single-tone RF signals with a frequency interval of 1 MHz between two neighboring signals. Figure 7 shows the temporal waveforms from the FSL. In Fig. 7, the FTM pulses corresponding to the optical carriers are aligned in the time domain as the reference pulses. Through measuring the time interval between the reference pulse and the neighboring FTM pulse corresponding to the five optically-carried single-tone RF signals, respectively, the frequency of the input RF signals can be calculated by using Eq. (5). On this condition, the FTM pulses corresponding to the five optically-carried single-tone RF signals can be fitted by using a straight dashed line in the time-frequency plane as depicted in Fig. 7, where the slope coefficient and the root-mean-square error of the linear fitting are 2.496 ns/MHz and 19 ps, respectively. These results indicate that an excellent FTM linearity is achieved. It should be pointed out that the relatively weak pulse corresponds to the -1^st-order modulation sideband of the SUT, which can be eliminated by using single sideband modulation to load the SUT onto the three-tone optical carrier.

Fig. 7. Temporal waveforms from the FSL for separately analyzing five single-tone RF signals with a frequency interval of 1 MHz between two neighboring signals.

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In the experiment, the ASE noise of the FSL seeded by a three-tone optical carrier in a unit bandwidth is identical to that of the FSL injected by a single-tone optical carrier due to the same actual number of recirculation, leading to an identical SNR in these two situations. The unambiguous bandwidth of the spectrum analysis in the experiment is 5 MHz, which is limited by the FSR of the FSL. Through shortening the loop delay, the unambiguous bandwidth can be further enhanced. Especially, if the FSL is integrated on a single chip, a gigahertz-level unambiguous bandwidth can be expected. In fact, the analog input bandwidth of the FSL-based photonic real-time Fourier transform is dependent on the modulation bandwidth of the MZM used to load the SUT onto the OFC. Hence, one fascinating feature of the FSL-based photonic real-time Fourier transform is that it can work in the frequency range up to tens of gigahertz or even beyond 100 GHz. Although the unambiguous bandwidth is constraint by the FSR of the FSL due to the frequency-domain sampling effect, it can be circumvented by using multiple parallel FSLs with coprime FSRs as the method used in [24,25]. In addition, the frequency resolution can be further improved through using an OFC with more teeth. The electro-optic frequency comb is a promising candidate to match the proposed scheme [26,27].

4. Conclusion

In summary, we have proposed and demonstrated an FSL-based high-resolution photonic real-time Fourier transform scheme based on OFC injection. The OFC with its frequency interval equal to an integer multiple of the frequency shift and also the FSR of the FSL greatly increases the number of the effective signal replicas from the FSL, which breaks the limitation on the number of round trips induced by the gain saturation effect and the cumulative ASE noise in the FSL. Hence, the frequency resolution is largely improved. In the experiment, through using coherent three-tone optical carrier injection, the frequency resolution was enhanced from 60 kHz to 20 kHz. The proposed scheme is beneficial for realizing high-resolution real-time spectrum analysis, ultra-short optical pulse generation and broadband chirp waveform generation based on an FSL architecture.

Funding

National Natural Science Foundation of China (61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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High-resolution real-time Fourier transform based on optical frequency comb injected frequency shifting loop

Abstract

1. Introduction

2. Operation principle

3. Experimental results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (8)

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